I know that this sentence is grammatically correct, but I still can't find a suitable grammar rule that would explain why strong enough is used after the noun in this sentence.

Why don't we say "There's no strong enough headache, that a good coffee won't relieve"?

It doesn't look like a reduced relative clause that usually occur with past participles, such as "the people skilled in design" ("the people which are skilled in design"). We usually place such adjectives like possible, available, and few others after the noun. This is not the case.

Can anybody explain the word order to me? It would be even better if someone provides a proof from a grammar book.

I love googling on partial phrases. Try searching on the phrase "there is no * enough" and scroll through the list. You will come across wonderful variants like, there is no flag large enough, there is no grave deep enough, and there is no threatening star close enough amongst many others.
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Peter RowellJan 1 '12 at 17:40

I think strong enough that a good coffee won't relieve is all one adjective phrase modifying headache. And we tend to put long, complex adjective phrases like this after the noun - There is no strong-enough-that-a-good-coffee-won't-relieve headache is almost uninterpretable.
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alcasJan 1 '12 at 17:43

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Although I have to say the original sentence sounds a bit odd to me as well. I'd be much more likely to say There is no headache strong enough that a good coffee won't relieve it.
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alcasJan 1 '12 at 17:43

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I've just found the rule I need in "A comprehensive grammar of the English Language" by Randolph Quirk... Now it makes sense! Thanks for the answer
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DesertJan 1 '12 at 18:52

1 Answer
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Strong enough means so strong that S, where S represents some clause relating to the strength. This is one example of a broader pattern, where both infinitive and that complement Ss can occur:

Xbe (Adjenough)/(soAdj) that/toS.

This is an Equative construction (the strength of X is such that S is true), but it works by specifying a minimum value of S, so it can be used with a high minimum to signify excessive degrees, and therefore is often used to boast:

Their beards are so durn tough, it seems, the hair just grows in lumps.

from "Why Oh Why Did I Ever Leave Wyoming?" by the Vic Jergens band, ca 1948 (the next line is That's why they shave with broken glass and dynamite the stumps.)

The position of strong enough in this particular sentence is after headache, which is the antecedent of the relative clause which is strong enough.... The which is part gets zapped by the usual Whiz-deletion, leaving only the predicate adjective phrase and its complement. So it's really a clause reduced to a modifying phrase, and that's why it sits after its antecedent.

It can also occasionally occur before the antecedent, especially if it's quantified with enough -- but not if it's quantified with with so or too.